So it's 3, and its imaginary part, minus 4 in the imaginary direction.

The phantom limb they'll say, But doctor, the phantom limb is paralyzed.

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Maybe you can relieve the phantom pain, the phantom cramp.

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Then the y axis is the imaginary axis.

And this part right here is imaginary.

And you all know what a phantom limb is.

Now, why would a phantom limb be paralyzed?

You take the imaginary part, minus 2i, minus 2, right?

And it's imaginary part is twoi, so you go along the imaginary axis, or the i axis you could even say, two.

And notice, we're just switching the direction in the imaginary along the imaginary axis, when we take the conjugate of something.

The vertical axis you could call it the imaginary axis.

And this will be the new imaginary part.

And now what's our new imaginary part?

Its imaginary part is minus 4 right there.

And that times i, and that's the imaginary part.

That is me playing my imaginary piano.

So lets draw this on an argand diagram, so that's my real axis, this is my imaginary axis, this is my imaginary axis, and so, this is the real, this the imaginary, and if I wanted to represent z equals 1, it only has a real part.

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So we would add or subtract the same imaginary number.

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Now, (Laughter) what does that have to do with the placebo effect?

So the real part is a, the imaginary part is bi.

So, for example, we could call this the complex number z sub one.

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The prevalence of phantom pain after limb amputation ranged from 50 to 78 .

But if you change the form that you give the placebo in,

And if you want the ultimate in placebo, you've go to the needle.

If you started with this, and changed the sign of the imaginary part, you would get seven minus five i.

But what this does is it allows us to get rid of the imaginary part in the denominator.

In the last two years of the study, all participants were switched to placebo.

Four of the seven subjects succeeded in performing impossible movements of the phantom limb.

So you can't merge, when you're adding and subtracting, the real part to the imaginary part.

like subprime mortgages or Bernie Madoff's pyramid scheme, the devil is always in the details.

The placebo effect is one of the most fascinating things in the whole of medicine.

In fact, seven trials were conducted comparing reboxetine against a dummy placebo sugar pill.

When I multiply a complex number times this other number, which is very similar to it, but it's kind of the imaginary part, goes in the other direction.

For some time I have been interested in the placebo effect, which might seem like an odd thing for a magician to be interested in, unless you think of it in the terms that I do, which is, Something fake is believed in enough by somebody that it becomes something real.

It turns out that if you administer a placebo in the form of a white pill, that's like aspirin shaped it's just a round white pill it has some certain measurable effect.

And this has been demonstrated in a whole raft of fascinating studies comparing one kind of placebo against another.

Consider the placebo effect, in which a pill with no active substances brings about a response similar to a pill with a substance present.

The phantom eye syndrome (PES) is a phantom pain in the eye and visual hallucinations after the removal of an eye (enucleation, evisceration).

One of the things we've found was, about half the patients with phantom limbs claim that they can move the phantom.